When reading about warfare records in ancient China, I see a lot of references to "# of chariots" instead of "#k soldiers" or something like that. Do we know how big these armies actually were? As in, if a state sent "300 chariots to the battle", how big a force was that actually?

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Using the number of chariots to denote the size of an army is most often a Spring and Autumn practice. During this period, the traditional, ritualistic formal system of the Chou Dynasty degraded and gave way to ad hoc reforms. Part of these reforms is changes in military organisation.

The Chou dynasty prescribed extensive rules governing most aspects of social and political life within their empire. These applied both to the royal domains, as well as most Chou-aligned tribal states concentrated in the Central Plains.

This original military system called for 30 soldiers for each chariot. That included 3 heavy infantry riding on the chariot, 7 more on foot, and 20 light infantry. At this stage, warfare was still a nobility centric event, and the light infantry were more servants than soldiers. At the start of the Spring and Autumn period, despite severe prestige and territorial losses by the Imperial Chou court when it moved east, many states still followed these rulse (even though they no longer obeyed royal commands).

By mid to late Spring and Autumn, increased warfare have caused the main states to restructure their military service. When the Chou Dynasty supplanted the Shang Dynasty, it established what is effectively colonies across its new territories. These "garrisons" (so to speak) became tribal cores for the Chou vassal states, and their descendants the ruling class. As citizens (國人, who live in the capital and the suburbs), they participated in state politics and for obvious reasons service in the army was restricted to them.

As warfare expanded, by necessity the states all enlarged their manpower pool. This was done by conscripting the conquered servile class (野人, who lived away from the city and in the fields). Still, riding and controlling the all-important chariot was a skill requiring long training, and as such remained the province of the lower nobility. Thus, to make use of the enlarged manpower pool, the chariot military unit was expanded to 75 soldiers per chariot, plus an additional 25 men logistic section. All in all this included: